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Classical electromagnetism
From Wikipedia, the free encyclopedia
(Redirected from Classical electrodynamics)
Classical electromagnetism (or classical electrodynamics) is a
branch of theoretical physics that studies consequences of the
electromagnetic forces between electric charges and currents. It
provides an excellent description of electromagnetic phenomena
whenever the relevant length scales and field strengths are large
enough that quantum mechanical effects are negligible (see
quantum electrodynamics). Fundamental physical aspects of
classical electrodynamics are presented e.g. by Feynman,
Leighton and Sands,[1] Panofsky and Phillips,[2] and Jackson.[3]
The theory of electromagnetism was developed over the course of
the 19th century, most prominently by James Clerk Maxwell. For
a detailed historical account, consult Pauli,[4] Whittaker,[5] and
Pais.[6] See also History of optics, History of electromagnetism
and Maxwell's equations.
Ribarič and Šušteršič[7] considered a dozen open questions in the
current understanding of classical electrodynamics; to this end
they studied and cited about 240 references from 1903 to 1989.
The outstanding problem with classical electrodynamics, as stated
by Jackson,[3] is that we are able to obtain and study relevant
solutions of its basic equations only in two limiting cases: »... one
in which the sources of charges and currents are specified and the
resulting electromagnetic fields are calculated, and the other in
which external electromagnetic fields are specified and the
motion of charged particles or currents is calculated...
Occasionally, ..., the two problems are combined. But the
treatment is a stepwise one -- first the motion of the charged
particle in the external field is determined, neglecting the
emission of radiation; then the radiation is calculated from the
trajectory as a given source distribution. It is evident that this
manner of handling problems in electrodynamics can be of only
approximative validity.« As a consequence, we do not yet have
physical understanding of those electromechanical systems where
we cannot neglect the mutual interaction between electric charges
and currents, and the electromagnetic field emitted by them. In
spite of a century long effort, there is as yet no generally accepted
classical equation of motion for charged particles, as well as no
pertinent experimental data, cf.[8]
Contents
Electromagnetism
Electricity · Magnetism
Electrostatics
Electric charge · Coulomb's law ·
Electric field · Electric flux ·
Gauss's law · Electric potential ·
Electrostatic induction ·
Electric dipole moment ·
Polarization density
Magnetostatics
Ampère's law · Electric current ·
Magnetic field · Magnetization ·
Magnetic flux · Biot–Savart law ·
Magnetic dipole moment ·
Gauss's law for magnetism
Electrodynamics
Lorentz force law · emf ·
Electromagnetic induction ·
Faraday’s law · Lenz's law ·
Displacement current ·
Maxwell's equations · EM field ·
Electromagnetic radiation ·
Liénard–Wiechert potential ·
Maxwell tensor · Eddy current
Electrical Network
Electrical conduction ·
Electrical resistance · Capacitance ·
Inductance · Impedance ·
Resonant cavities · Waveguides
Covariant formulation
■ 1 Lorentz force
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Classical electromagnetism - Wikipedia, the free encyclopedia
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2 The electric field E
3 Electromagnetic waves
4 General field equations
5 Models
6 See also
7 References
8 External links
Lorentz force
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Electromagnetic tensor ·
EM Stress-energy tensor ·
Four-current ·
Electromagnetic four-potential
Scientists
Ampère · Coulomb · Faraday ·
Gauss · Heaviside · Henry · Hertz ·
Lorentz · Maxwell · Tesla · Volta ·
Weber · Ørsted
Main article: Lorentz force
The electromagnetic field exerts the following force (often called the Lorentz force) on charged
particles:
where all boldfaced quantities are vectors: F is the force that a charge q experiences, E is the electric
field at the location of the charge, v is the velocity of the charge, B is the magnetic field at the location
of the charge.
The above equation illustrates that the Lorentz force is the sum of two vectors. One is the cross product
of the velocity and magnetic field vectors. Based on the properties of the cross product, this produces a
vector that is perpendicular to both the velocity and magnetic field vectors. The other vector is in the
same direction as the electric field. The sum of these two vectors is the Lorentz force.
Therefore, in the absence of a magnetic field, the force is in the direction of the electric field, and the
magnitude of the force is dependent on the value of the charge and the intensity of the electric field. In
the absence of an electric field, the force is perpendicular to the velocity of the particle and the direction
of the magnetic field. If both electric and magnetic fields are present, the Lorentz force is the sum of
both of these vectors.
The electric field E
Main article: Electric field
The electric field E is defined such that, on a stationary charge:
where q0 is what is known as a test charge. The size of the charge doesn't really matter, as long as it is
small enough not to influence the electric field by its mere presence. What is plain from this definition,
though, is that the unit of E is N/C, or newtons per coulomb. This unit is equal to V/m (volts per meter),
see below.
The above definition seems a little bit circular but, in electrostatics, where charges are not moving,
Coulomb's law works fine. The result is:
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where n is the number of charges, qi is the amount of charge associated with the ith charge, ri is the
position of the ith charge, r is the position where the electric field is being determined, and ε0 is the
electric constant.
Note: the above is just Coulomb's law, divided by q1, adding up multiple charges.
If the field is instead produced by a continuous distribution of charges, the summation becomes an
integral:
where ρ(r) is the charge density as a function of position, is the unit vector pointing from dV to the
point in space E is being calculated at, and r is the distance from the point E is being calculated at to the
point charge.
Both of the above equations are cumbersome, especially if one wants to calculate E as a function of
position. There is, however, a scalar function called the electrical potential that can help. Electric
potential, also called voltage (the units for which are the volt), which is defined by the line integral
where φE is the electric potential, and C is the path over which the integral is being taken.
Unfortunately, this definition has a caveat. From Maxwell's equations, it is clear that ∇ × E is not
always zero, and hence the scalar potential alone is insufficient to define the electric field exactly. As a
result, one must resort to adding a correction factor, which is generally done by subtracting the time
derivative of the A vector potential described below. Whenever the charges are quasistatic, however,
this condition will be essentially met, so there will be few problems.
From the definition of charge, one can easily show that the electric potential of a point charge as a
function of position is:
where q is the point charge's charge, r is the position, and rq is the position of the point charge. The
potential for a general distribution of charge ends up being:
where ρ(r) is the charge density as a function of position, and r is the distance from the volume element
dV.
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Note well that φ is a scalar, which means that it will add to other potential fields as a scalar. This makes
it relatively easy to break complex problems down in to simple parts and add their potentials. Taking the
definition of φ backwards, we see that the electric field is just the negative gradient (the del operator) of
the potential. Or:
From this formula it is clear that E can be expressed in V/m (volts per meter).
Electromagnetic waves
Main article: Electromagnetic waves
A changing electromagnetic field propagates away from its origin in the form of a wave. These waves
travel in vacuum at the speed of light and exist in a wide spectrum of wavelengths. Examples of the
dynamic fields of electromagnetic radiation (in order of increasing frequency): radio waves,
microwaves, light (infrared, visible light and ultraviolet), x-rays and gamma rays. In the field of particle
physics this electromagnetic radiation is the manifestation of the electromagnetic interaction between
charged particles.
General field equations
Main articles: Jefimenko's equations and Liénard-Wiechert Potentials
As simple and satisfying as Coulomb's equation may be, it is not entirely correct in the context of
classical electromagnetism. Problems arise because changes in charge distributions require a non-zero
amount of time to be "felt" elsewhere (required by special relativity).
For the fields of general charge distributions, the retarded potentials can be computed and differentiated
accordingly to yield Jefimenko's Equations.
Retarded potentials can also be derived for point charges, and the equations are known as the LiénardWiechert potentials. The scalar potential is:
where q is the point charge's charge and r is the position. rq and vq are the position and velocity of the
charge, respectively, as a function of retarded time. The vector potential is similar:
These can then be differentiated accordingly to obtain the complete field equations for a moving point
particle.
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Models
A branch of classical electromagnetisms such as optics, electrical and electronic engineering consist of a
collection of relevant mathematical models of different degree of simplification and idealization to
enhance our understanding of the specific electrodynamics phenomena, cf.[9] An electrodynamics
phenomenon is determined by the particular fields, specific densities of electric charges and currents,
and the particular transmission medium. Since there are infinitely many of them, in modeling there is a
need for some typical, representative
(a) electrical charges and currents, e.g. moving pointlike charges and electric and magnetic
dipoles,[10] electric currents in a conductor etc;
(b) electromagnetic fields, e.g. voltages, the Liénard-Wiechert potentials, the monochromatic
plane waves , optical rays; radio waves, microwaves, infrared radiation, visible light, ultraviolet
radiation, X-rays , gamma rays etc;
(c) transmission media, e.g. electronic components, antennas, electromagnetic waveguides, flat
mirrors, mirrors with curved surfaces convex lenses, concave lenses; resistors, inductors,
capacitors, switches; wires, electric and optical cables, transmission lines, integrated circuits etc;
which all have only few variable characteristics.
See also
■ Quantum electrodynamics
■ Wheeler-Feynman absorber theory
References
1. ^ Feynman, R.P., R.B. Leighton, and M. Sands, 1965, The Feynman Lectures on Physics, Vol. II: the
Electromagnetic Field, Addison-Wesley, Reading, Mass.
2. ^ Panofsky, W.K., and M. Phillips, 1969, Classical Electricity and Magnetism, 2nd edition, AddisonWesley, Reading, Mass.
3. ^ a b Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). New York: Wiley. ISBN 0-471-30932-X.
4. ^ Pauli, W., 1958, Theory of Relativity, Pergamon, London
5. ^ Whittaker, E.T., 1960, History of the Theories of the Aether and Electricity, Harper Torchbooks, New
York.
6. ^ Pais, A., 1983, »Subtle is the Lord...«; the Science and Life of Albert Einstein, Oxford University Press,
Oxford
7. ^ Ribarič, M., and L. Šušteršič, 1990, Conservation Laws and Open Questions of Classical Electrodynamics,
World Scientific, Singapore
8. ^ M. Ribarič and L. Šušteršič, Improvement on the Lorentz-Abraham-Dirac equation, arxiv:1011.1805
(http://arxiv.org/abs/1011.1805)
9. ^ Peierls, Rudolf. Model-making in physics
(http://www.informaworld.com/smpp/content~content=a752582770~db=all~order=page) , Contemporary
Physics, Volume 21 (1), January 1980, 3-17.
10. ^ [ Ribarič M. and Šušteršič L. Moving pointlike charges and electric and magnetic dipoles, AM.J.Phys.60
(6),June 1992 ]
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External links
■ Electromagnetic Field Theory (http://www.plasma.uu.se/CED/Book/EMFT_Book.pdf) by Bo
Thidé
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Categories: Electromagnetism | Electrodynamics
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